


Amplitude, Frequency, and Phase

by kusuriurikun



Series: Shibuya Continuum [5]
Category: Subarashiki Kono Sekai | The World Ends With You
Genre: Canonical Character Death, Character Study, Gen, Math and Science Metaphors, Mathematics, Noise Forms, Readers are forewarned I let Sho say "fuck", Shooting Guns, Sparring, There may be Weebers within, fun with attempted assassination of a local demigod
Language: English
Status: Completed
Published: 2020-10-01
Updated: 2020-10-01
Packaged: 2021-03-08 00:27:24
Rating: Teen And Up Audiences
Warnings: Graphic Depictions Of Violence, Major Character Death, No Archive Warnings Apply
Chapters: 2
Words: 2,982
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/26716732
Author URL: https://archiveofourown.org/users/kusuriurikun/pseuds/kusuriurikun
Summary: Everything is part of a mathematical function, Sho Minamimoto thinks.Axiomatic shifts are, however, sometimes forced when experiencing variablescompletelyoutside your particular system of mathematics...and you find yourselfcompletelyout of your vector.As usual, due to heavy use of mathematical and engineering metaphors, notes included in an appendix at the end(Trigger warnings for pepperbox revolvers and canonical character death.)
Relationships: Hanekoma Sanae & Minamimoto Sho
Series: Shibuya Continuum [5]
Series URL: https://archiveofourown.org/series/1902298
Comments: 2
Kudos: 9





	1. Main Story

It is honestly surprising--and really, comforting to Sho, when he thinks on it--just how much the nature of reality in the UG, and its interaction with the RG, is beautifully mathematical.

The shift from the usual more human Reaper form to the Noise-forms, he muses, is a function of a shift in phase modulation--a rotation a few degrees one way or another in the essential sine-wave brings a shift in the sine and cosine and tangent transferring variables from the real to the imaginary in an equation as simple--and expressible--as _Z=R+iX_ or _e iωt=cos ωt+i(sin ω_ _t)_. 

The shift from the UG to the RG is itself a function of frequency modulation--one of the things Reapers learn early on, though probably not to the extent and the methods Sho uses; most of the Reapers used a meditation or centering method, Sho actively calculates the sine of his own function until the shift _down_ to the new frame of reference.

(It’s been the method that always worked best for him, at any rate. The world is made of numbers, after all; he’d made an axiomatic choice early on in regards to finding THAT solution.)

* * *

At one point, the eccentric barista of the WildKat Cafe gets in touch, and sends a note offering him (as he put it) some education on refining the complex function calling forth his own Noise and using it Well.

So Sho shows up at 1:41am (he appreciated the fun Hanekoma had with _times_ for experiments like these), and Hanekoma calls him in. 

Tables and chairs have been cleared from the front area, the espresso machine secured. It’s now sparring practice time.

“Let us begin.” Hanekoma calls the phase shift, and calls out the great singing panther that is _his_ beast even as the singing lion is Sho’s.

Sho comes in, fierce, feral, the cat--the _lion_ \--inward even as he is still human outward, still mostly in the real part of the complex domain, riding the adrenalin…

...and finds himself facefirst on the cafe’s floor.

Adrenalin goes to rage, as lion snaps at the other cat which veritably _dances_ away and laughs, at least as much as large pantherines can laugh so much as _chuff_.

The shift--now with the larger sum of his personal matrix in the imaginary--a shift of self, a blur of fur and claws and electricity as his inner Noise takes form and the singing lion is manifested without as well as within and a shift of phase happens as opposed to frequency.

Again a strike. Again the panther leaps just out of reach.

 _What the factor is this hectopascal doing?_ Sho thinks in a rage. 

Hears laughter...though less heard than felt. Like if somehow sound itself was turned in phase ninety degrees like a polarized film.

_Fuck._

Realises what’s up. Fucker’s not even breaking a sweat, Hanekoma isn’t.

Gets a sense out of SOMEwhere, leaps JUST out of the way before the leg sweep...to find himself almost running into Noise-paw-boot-in-face as _what-was-and-is-Hanekoma_ has somehow fucking _somersaulted_ to the other side in a way that should not be _possible._

Except it’s happening, and almost by instinct Sho finds himself in another, slightly different, coordinate of the plane. 

Grins. _Hah. So zetta slow…_

Has that grin wiped off his face approximately 1.414 seconds later by the sense ( _the sense in the back of the neck the spine the tail_ ) that there is Incoming, this time tries to meet it...again to have the panther leap again out of the way.

_The motherfactoring hell? Did that function just...project itself **wings**?_

A paw comes out of nowhere, sends him sprawling. He’s getting wiped out now, this particular amplitude degenerating. The matrix of Sho Minamimoto goes back to the dominant sum being in the real, the lion becoming the wild-child with fangs just a bit too long to be entirely human.

A thousand lights, and then Sho perceives a hand.

A hand up. Lifting him to a position he can sit up.

(He is somewhat puzzled the WildKat somehow looks none the worse for wear.)

Hanekoma also has dropped his own phase shift, his own transformation in the plane; where there was the blur of fur and feathers and electricity there’s the barista and artist and--apparently--a minor master of capoeira, or at least something _resembling_ it.

“Not bad, kid. But I can show you how to do better.”

* * *

As it turns out, Sho is a _remarkably_ apt pupil indeed, and there are many more things to learn.

One of the first things Sho learns is that Hanekoma’s back “office” is more like a _studio_ in the back of the WildKat Cafe, complete with paints and concept sketches...and also a surprisingly stocked mini-library, which somehow has its selection ever-changing of old books that somehow seem to be on the _exact_ sorts of things where he has a particular curiosity.

When not sparring in the Cafe itself at night, or (increasingly) on the roof of the Pork City complex, Sho finds himself in study; not only on the books he has in the back, but putting these observations into practice.

Sho has noticed that the amplitude of the sine function of the living is of a decidedly different nature to that of Players, and Noise different still; the amplitude of the background noise approaches zero, the generated Noise being primarily a function of frequency, the living having their own amplitude at higher or lower volume depending on their own emotional state.

And he knows one of the annoyances (relatively speaking) of the RG is that there is a bound on phase; because of its nature as a lower-vibrational plane, there are limits to wave functions, and--just as a three-dimensional function collapses to a line or a 2d shape on an XY plot--so the essential function that allows the UG to be...well... _the UG_ , the background noise and phase shifts collapses to FM, like the generators in old Yamaha synthesizers.

(Sho had asked Hanekoma on why things worked differently and why Noise was impossible in the RG. Hanekoma was the first to actually explain this to him in a way that made Sense--the imaginary axis was _very_ limited, almost to eliminate that particular dimension of freedom at all. He’d also taught him some interesting things regarding signal and noise along the way, between sparring and learning to modulate that phase shift _just_ so to greatest precision and discussions on the Management over macchiato and mathematics.)

Occasionally Hanekoma even lets some things slip on the differences between the Noise that avoided those with a Pact but cropped up via the everyday perturbations of the complex plane, and Pin-bound Noise that was pulled through and set and Tamed via formula and a Reaper’s will...and _other_ Noise that was not bound to either function.

Hanekoma is always _exceedingly_ careful when he talks about that third kind...almost as if it were something forbidden to even contemplate, much less discuss. At one point even asks Sho if he actually _needs_ his Leo pin to shift or to call forth Noise.

Sho realizes he does _not_ , and realizes at least some of the power and danger inherent in the taboo things Hanekoma talks about only in the quietest of hush-hush tones, and how it relates to how _he_ makes things work. 

The strange black Noise he calls up (and which has touched his hand) and his Beast are unbound functions, he understands.

And all this information, this mathematical Theory of Everything that Sho keeps safely hidden in notes to himself in the trash-henge he now affectionately refers to as the Memory Heap when he bothers to even write it down to begin with (the most secret things ciphered, if written at all)...well...it becomes very useful when he learns that the Composer is running around in the RG, and becomes more useful still when he receives the gift of the pepperbox revolver deposited next to his own mathematical grimoire.

 _This...well, **this** is certainly a happy rebalancing of the equation_, Sho thinks to himself on hearing the news. Less variables means it’s easier to derive a desired solution.

And that particular variable of the Composer being able to set his own axioms that everyone else was bound to as rules of the local algebra of the UG was an _annoying_ one indeed.

* * *

The dampening function that is the downshift in frequency is always rough getting used to after so long in the UG with its own additional degree of freedom; much like in one of his favourite books _Flatland_ how the 3-dimensional beings don’t really fit in 2D space save as in shadow, the RG and the fact that its imaginary dimension is very much _limited_ does put a damper on things.

That said, it also can give more of an axis of freedom, Sho notes.

No strange rumors, and harder to track, but--so he thinks--the Composer is just as bound by this particular limit on this function as he is.

There are only so many places the Composer can go, and Shibuya isn’t infinite, so it’s a function of time before that fractal bastard is run to ground.

Sho walks out of Ramen Don, looks around. Crosses in front of the 104 (except here in the RG it’s the _109_ ). 

A passerby stares, as if she can see something about Sho out of place. She is rewarded with a grin of all-too-sharp teeth, and nearly trips on the sidewalk.

* * *

He actually _finds_ the Fractal Bastard at one of the places in Shibuya where the Real and the Imaginary overlap in an almost perfect 1+ _i_ function, CAT’s mural.

(Sho knows it’s designed that way. That most beautiful of strange attractors.)

He’s somehow not shocked to see Joshua there at the mural.

He _is_ distinctly shocked to see Joshua pointing a gun at some kid who’s just minding his own business, zetta _oblivious_ as he’s taking in the graffiti and whatever poppy garbage he’s got going through those headphones of his.

Like Joshua’s gonna _shoot_ the headphones kid. 

This...Sho does not claim to be an expert on the rules of the higher Game, but he does know that in general specifically adding Players is... _discouraged,_ to say the least. It’s not within the game theory; it’s not considered _sporting._

(If it were _that_ easy, Reapers would be selecting future Players left and right, after all.)

The man in black--hoping that the man in white is distracted--pulls out his _own_ revolver, at a rate that would be surprisingly quick by what folks in the RG would be used to, but is _so zetta slow_ to his own senses.

The headphones kid looks. He’s fucking terrified. There’s a part of Sho that wants to scream at the kid “ _run_ , you factoring hectopascal, you’re out of your _vector,_ _quit being an unbalanced variable in my equation you monomial!_ ”...not that it’d matter. Headphones Kid is frozen, like a deer in the headlights.

Like _Sho_ was on that fateful day when that fucking garbage truck plowed into him at 40 kilometers per hour.

_...no matter. Can’t let the extra variable interfere, it’ll get balanced later…_

Sho fires. It’s the loudest thing he’s ever heard, and he’s knocked nearly deaf, and by some minor miracle he doesn’t hit the kid…

...but _he doesn’t hit Joshua, either._

The Fractal Bastard smirks, and fires back, and a white-hot lance of pain fires through Sho’s right shoulder...which he grabs at, trying to keep his frequency _down_ as he aims and fires again.

And finds the bullets utterly fail to hit the Composer.

Not because he misses. Because the bullets never quite reach their trajectory.

_Because the fucking bullets **stop in midair** , hang, and then fall. _

Sho looks at this dumbfounded; this is literally something that should not happen.

This was fucking _impossible._ This is something that should _not be doable_ based on what he knows of the limits of the imaginary line in the RG, and the particular algebra and axioms it operates under. 

It should be every bit as impossible as dividing by zero. It’s something that should be literally an undefined _function_.

This is as zetta incomprehensible as actually expecting the ratio of the radius and the circumference of a circle to measure exactly 3 (instead of its first few digits of 3.14159265358979323846264338327 and onwards to infinity)...and yet he _sees_ this, despite the fact it is as likely as adding 3+1 and actually getting a _rabbit_ dressed in finest Lapin Angelique fashion.

Sho realizes that in this particular calculation...he’s _way_ out of his vector himself. 

Whispers to himself _I blew it_ in a mass of pain and shock and frank disbelief, and finally loses his composure and starts to fade back upwards into the UG.

As he crosses the transition boundary and the imaginary line explodes to infinity, he sees--hears--Joshua calmly walk up and shoot the kid with the headphones (who is still terrified) point blank.

And all Sho can do as his own wave function collapses is growl and curse as he hobbles back to the WildKat Cafe to lick his wounds.

This was impossible. What the Composer had done should literally be _impossible_. This was the equivalent of factoring the square root of four and getting _toast._

This...was going to require a _major_ recalculation of his plans.


	2. Appendix: Author's notes and terminology

**Summary for the Chapter:**

> (Or “oh dear gods, the author has put so much damned math and physics and obscure terminology in that a glossary is actually necessary to understand the fic!” Probably a sign of Bad Storytelling, really, but…)

The formula I mention at the start is actually the classical formula (in two different formats) for [calculation of complex impedance](http://hyperphysics.phy-astr.gsu.edu/hbase/electric/impcom.html)\--say, in an electronics circuit or an antenna transmission line--and to map out impedance and reactance in a polar plot in the form of sines, cosines, and tangents. (I admittedly did write the formula SLIGHTLY differently than is the convention in engineering circles--using _i_ rather than _j_ for notation of the imaginary component--to tie in more with Sho’s general Obsession with complex equations and other things I’ve written before re his obsession with the Euler identity and complex equations in general.

The three methods of modulation called out here--[amplitude modulation](https://en.wikipedia.org/wiki/Amplitude_modulation), [frequency modulation](https://en.wikipedia.org/wiki/Frequency_modulation), and [phase modulation](https://en.wikipedia.org/wiki/Phase_modulation)\--also tie in pretty heavily, and the specific choice for phase modulation representing Noise forms (and really, Noise in general, but we’ll get to that in a future fic…) actually has both stylistic and actual engineering-related reasons. (Frequency modulation and phase modulation both specifically modulate an angle at which a wave-form oscillates, whereas amplitude modulation effectively is based on the signal level or “height” of a waveform. Pretty much anyone who’s heard a radio station is familiar with AM and FM radio; the specific addition of phase modulation or “PM” does require a bit of explanation, though.

Phase modulation is actually somewhat related to FM in that both specifically modulate a signal based on the interaction of a carrier wave and a modulator (in other words, itself playing with sines and cosines and tangents). PM (versus FM) tends to do this functionally at a “right angle”; very interestingly, PM is often used to _generate_ FM and in a lot of circuit design is treated as functionally equivalent, and it works really nicely (IMHO) aesthetically as a way to implement (at least mathematically speaking) how Sho really does tend to see how Noise works--and also gives the author a Neat Out to explain why the hell you (normally) don’t have Noise popping up in the RG on a regular basis.

[ _Flatland_](https://en.wikipedia.org/wiki/Flatland) (which, of note, is available in full via the [Open Library](https://openlibrary.org/works/OL118420W), among other sources; it’s long since gone into the public domain) is a wonderfully odd duck of a book, in that it’s simultaneously a beloved book by mathematicians and mathgeeks (as it plays around _extensively_ with the concept of topological dimensions!) _and_ was also originally meant as a rather biting parody/commentary on Victorian society (and especially classism) in 1880s England. It’d certainly be something Sho would appreciate on the maths level and--once clued in--probably the political-farce level as well.

As amazing as it sounds, the guns depicted in TWEWY are _not_ the equivalent of the Disco Guns in the infamous _Gundam Seed_ Toonami dub, but it’s an actual type of early revolver--namely the [pepperbox revolver](https://en.wikipedia.org/wiki/Pepper-box), with the TWEWY pepperboxes being in particular apparently (loosely) based on the J. Chaineux Pinfire or possibly a Santander pepperbox or one of the many clones thereof. (This particular style of gun was popular as a concealed weapon, especially in the days before semiautomatics.)

* * *

I apologise to my readers in advance that literally every fic I write regarding Sho Minamimoto includes a bonus college-level engineering maths course. This is effectively what happens when an actual engineer (who also is a licensed Extra class amateur radio operator and has had to engage in engineering math, including calculation of antenna impedance and resistance via Smith charts) who was an extremely maths-obsessed dork of a kid (and is still a maths-obsessed dork of an _adult_ who wishes I knew more calculus than I actually do) actually vibes harder on an Extremely Relatable Feral Mathgeek than is reasonable.

(Luka/anguiscantus @ Twitter--if you are wondering if Sho would want to be an engineering student, the answer is _zetta yes_.)

I’d like to thank Willow in particular for the roughing-out/mini-story that was the origin of the sparring match in this particular fic, Seb and Jaynis and AliaMiel and Siletrea for ideas we’ve bounced back and forth regarding headcanons in general here to make this a bit more cohesive, and you for reading this trash-heap of a fic :D

The reader is forewarned the future stories are going to progress to being very dark. Intentionally so, in fact…but we’ll get to that in a function of time.


End file.
